The chance of getting all of the cards right on your first guess is 1/52!, but if you get all but two, then you actually know the position of the last two (it's the opposite of the way you had them!). Three doesn't work, because you only know one of the ways that the three aren't arranged, but there are more than one other ways.
So what is the chance that you know all but two? If it was only the first two, then it would be a chance of 1/52! as well, because there is only one permutation that has them switched. Because it can be any two, and there are 52*51 possible positions for the two to be placed in the list, it is 51*52/52! = 1/49!.
But we need to remember the chance that we got them all right to begin with, so the final answer is:
(52*51 + 1)/52! = 2653/52!